Large deviations results for some stochastic partial differential equations

In this thesis we consider two reaction-diffusion equations with small random perturbations.;In the first problem the perturbation is a Brownian sheet and the solution has paths in C(0,T;;In the second problem we consider a nonlinear equation in two space dimensions perturbed by a certain Wiener process as a good approximation to the same (solution-less) problem perturbed by a cylindrical Wiener process. We show that the energy of a related control problem for which an explicit formula is available may be used as an approximation for the large deviations principle of the invariant measure of the equation as the perturbation gets close to the cylindrical Wiener process in a weak sense that we describe.